Finite Group Databasebeta

← #34113202

Subgroups — #34113202

Order 20096.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758008C211yesmaximal, characteristic, complemented
2#23757665C211yesmaximal, characteristic, complemented
3#23757663C211yesmaximal, characteristic
4#23757655C211yesmaximal, characteristic, complemented
5#23757495C211yesmaximal, characteristic, complemented
6#23757492C211yesmaximal, characteristic, complemented
7#23757488C211yesmaximal, characteristic, complemented
8#23757485C211yesmaximal, characteristic, complemented
9#23757475C211yesmaximal, characteristic, complemented
10#23757473C211yesmaximal, characteristic, complemented
11#23757404C211yesmaximal, characteristic, complemented
12#23757383C211yesmaximal, characteristic, complemented
13#23757378C211yesmaximal, characteristic, complemented
14C4×#14578941C211yesmaximal, characteristic, complemented
15#556×C157C211yesmaximal, characteristic, complemented
16C2×#14578944C2211yescharacteristic
17C2×#14578944C2211yescharacteristic
18C2×#14578942C2211yescharacteristic
19C2×#14578941C2211yescharacteristic
20C2×#14578941C2211yescharacteristic
21C2×#14578941C2211yescharacteristic
22C2×#14578941C2211yescharacteristic
23C2×#14578939C2211yescharacteristic
24C2×#14578939C2211yescharacteristic
25#19097768C2211yescharacteristic
26#19097760—21no—
27#19097759C2211yescharacteristic
28#19097759C2211yescharacteristic
29#19097756—21no—
30#19097756—21no—
31C4×Dic314C2211yescharacteristic
32C4×D628C2211yescharacteristic
33C4×D628C2211yescharacteristic
34#19097733—21no—
35#19097702C2211yescharacteristic
36C4⋊#11929434C2211yescharacteristic
37C4⋊#11929434C2211yescharacteristic
38C4⋊#11929434C2211yescharacteristic
39#19097695C2211yescharacteristic
40#19097695C2211yescharacteristic
41#19097694C2211yescharacteristic
42#19097692C2211yescharacteristic
43#19097691C2211yescharacteristic
44#19097691C2211yescharacteristic
45#19097682C2211yescharacteristic
46#19097682C2211yescharacteristic
47C4×#11929434C2211yescharacteristic
48C4⋊Q8×C157C2211yescharacteristic
49#126×C157—21no—
50#126×C157—21no—
51#124×C157C2211yescharacteristic
52#124×C157C2211yescharacteristic
53C22⋊Q8×C157—21no—
54C4×Q8×C157C2211yescharacteristic
55C4×Q8×C157C2211yescharacteristic
56C4×Q8×C157C2211yescharacteristic
57C4×D4×C157C2211yescharacteristic
58#117×C157—21no—
59C2×Dic314C2311yescharacteristic
60C2×D628—21no—
61C2×D628—21no—
62C2×D628C2311yescharacteristic
63C2×D628C2311yescharacteristic
64C2×C4×D157—21no—
65C2×C4×D157—21no—
66#14578944—22no—
67#14578944—41no—
68#14578942—41no—
69#14578941—22no—
70#14578941—41no—
71#14578941—41no—
72#14578941—41no—
73#14578939—22no—
74#14578939—41no—
75C2×#11929434C2311yescharacteristic
76C2×#11929434C2311yescharacteristic
77C2×#11929434C2311yescharacteristic
78C2×#11929434C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81#14578931—21no—
82#14578931—21no—
83#14578931—21no—
84#14578931—21no—
85#14578931—21no—
86#14578931—21no—
87C4×Dic157—21no—
88C2×Q8×C157D411yescharacteristic
89C2×Q8×C157C2311yescharacteristic
90C2×Q8×C157C2311yescharacteristic
91C2×D4×C157C2311yescharacteristic
92C22×C4×C157—21no—
93C4⋊C4×C157—21no—
94C4⋊C4×C157—21no—
95C4⋊C4×C157—21no—
96C4⋊C4×C157—21no—
97C4⋊C4×C157D411yescharacteristic
98C4⋊C4×C157D411yescharacteristic
99C4⋊C4×C157D411yescharacteristic
100C4⋊C4×C157C2311yescharacteristic
101C22⋊C4×C157—21no—
102C22⋊C4×C157—21no—
103C22⋊C4×C157—21no—
104C22⋊C4×C157—21no—
105C22⋊C4×C157—21no—
106C42×C157—21no—
107C42×C157—21no—
108C42×C157C2311yescharacteristic
109C42×C157C2311yescharacteristic
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157—21no—
113Dic314—21no—
114Dic314—21no—
115D628—21no—
116D628—21no—
117D628—21no—
118D628—21no—
119D628—42no—
120D628—81no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C4×D157—81no—
124C4×D157—81no—
125#11929434—21no—
126#11929434—21no—
127#11929434—21no—
128#11929434#4112yes—
129C23×C157—21no—
130Q8×C157—21no—
131Q8×C157—21no—
132Q8×C157—21no—
133Q8×C157—21no—
134Q8×C157—41no—
135D4×C157—21no—
136D4×C157—21no—
137C2×C4×C157—21no—
138C2×C4×C157—21no—
139C2×C4×C157—21no—
140C2×C4×C157—42no—
141C2×C4×C157C2×D411yescharacteristic
142C2×C4×C157C2×D411yescharacteristic
143C2×C4×C157C2×D411yescharacteristic
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2×D411yescharacteristic
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2411yescharacteristic
148C22×C157—41no—
149C22×C157—41no—
150C22×C157—41no—
151C22×C157C22×D411yescharacteristic
152C2×D157—41no—
153C2×D157—41no—
154C2×D157—41no—
155C2×D157—41no—
156C2×D157—41no—
157C2×D157—41no—
158C4×C157—21no—
159C4×C157—21no—
160C4×C157—21no—
161C4×C157—21no—
162C4×C157—21no—
163C4×C157—21no—
164C4×C157—41no—
165C4×C157—41no—
166C4×C157—41no—
167C4×C157C2×#4111yescharacteristic
168C4×C157H(5,𝔽2)11yescharacteristic
169Dic157—41no—
170Dic157—41no—
171C2×C157—81no—
172C2×C157#55011yescharacteristic
173C2×C157#57511yescharacteristic
174C2×C157#57711yescharacteristic
175D157—81no—
176D157—81no—
177C157#340111yescharacteristic, complemented, Sylow
178#3401—1571nomaximal, Sylow
179#559—1571no—
180#556D15711yescharacteristic, complemented
181#497—1571no—
182#496—1571no—
183#495—1571no—
184#464—1571no—
185#464—1571no—
186#463—1571no—
187#463—1571no—
188#461—1571no—
189#460—1571no—
190#446—1571no—
191#442—1571no—
192#441—1571no—
193C4×SD16—1571no—
194C2×Q16—1571no—
195C2×Q16—1571no—
196C2×SD16—1571no—
197C2×SD16—1571no—
198C2×SD16—1571no—
199C2×SD16—1571no—
200C2×SD16—1571no—
201C2×D8—1571no—
202C2×D8—1571no—
203C4⋊Q8C2×D15711yescharacteristic
204#127—1571no—
205#126—21no—
206#126—21no—
207#124—3141no—
208#124—1571no—
209#124—1571no—
210#124C2×D15711yescharacteristic
211#124C2×D15711yescharacteristic
212#123—3141no—
213#123—3141no—
214C22⋊Q8—21no—
215C4×Q8—1571no—
216C4×Q8C2×D15711yescharacteristic
217C4×Q8C2×D15711yescharacteristic
218C4×Q8C2×D15711yescharacteristic
219C4×D4—3141no—
220C4×D4—1571no—
221C4×D4—1571no—
222C4×D4C2×D15711yescharacteristic
223#117—21no—
224#106—1571no—
225C4⋊C8—1571no—
226C4⋊C8—1571no—
227C4⋊C8—1571no—
228#103—1571no—
229#103—1571no—
230#103—1571no—
231#102—1571no—
232#102—1571no—
233#102—1571no—
234#97—1571no—
235#97—1571no—
236C4×C8—1571no—
237C2×Q8—1571no—
238C2×Q8C22⋊D15711yescharacteristic
239C2×Q8C22×D15711yescharacteristic
240C2×Q8C22×D15711yescharacteristic
241C2×D4—3141no—
242C2×D4—3141no—
243C2×D4—1571no—
244C2×D4—1571no—
245C2×D4C22×D15711yescharacteristic
246C22×C4—21no—
247C22×C4—3141no—
248C22×C4—3141no—
249Q16—3142no—
250Q16—6281no—
251SD16—3142no—
252SD16—6281no—
253SD16—6281no—
254SD16—6281no—
255SD16—6281no—
256D8—3142no—
257D8—6281no—
258C2×C8—1571no—
259C2×C8—1571no—
260C2×C8—1571no—
261C2×C8—1571no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—3141no—
267C4⋊C4—1571no—
268C4⋊C4C22⋊D15711yescharacteristic
269C4⋊C4C22⋊D15711yescharacteristic
270C4⋊C4C22⋊D15711yescharacteristic
271C4⋊C4C22×D15711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3141no—
278C22⋊C4—3141no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C42—21no—
284C42—21no—
285C42—3141no—
286C42C22×D15711yescharacteristic
287C42C22×D15711yescharacteristic
288C42C22×D15711yescharacteristic
289C23—21no—
290C23—3141no—
291C23—3141no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—41no—
297Q8—3141no—
298Q8—3141no—
299D4—21no—
300D4—21no—
301D4—3141no—
302D4—3141no—
303D4—3141no—
304D4—3141no—
305D4—6282no—
306D4—12561no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—12561no—
314C2×C4—12561no—
315C2×C4C2×C22⋊D15711yescharacteristic
316C2×C4C2×C22⋊D15711yescharacteristic
317C2×C4C2×C22⋊D15711yescharacteristic
318C2×C4C2×C22⋊D15711yescharacteristic
319C2×C4C2×C22⋊D15711yescharacteristic
320C2×C4C2×C22⋊D15711yescharacteristic
321C2×C4C23×D15711yescharacteristic
322C8—3141no—
323C8—3141no—
324C8—3141no—
325C8—1572no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—6281no—
330C22—6281no—
331C22—6281no—
332C22—6281no—
333C22—6281no—
334C22—6281no—
335C22C22×C22⋊D15711yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—6281no—
346C4—6281no—
347C4#41×D15711yescharacteristic
348C4#1909781711yescharacteristic
349C2—81no—
350C2—12561no—
351C2—12561no—
352C2#2375794211yescharacteristic
353C2#2375807111yescharacteristic
354C2#2375808111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#556×C157C2
socleC22×C157C22×D4
prime radical#23757663C2
hypercenter#556D157
hypocenterC157#3401
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #23758008C2semidirect
2. #23757665C2semidirect
4. #23757655C2semidirect
5. #23757495C2semidirect
6. #23757492C2semidirect
7. #23757488C2semidirect
8. #23757485C2semidirect
9. #23757475C2semidirect
10. #23757473C2semidirect
11. #23757404C2semidirect
12. #23757383C2semidirect
13. #23757378C2semidirect
14. C4×#14578941C2semidirect
15. #556×C157C2semidirect
177. C157#3401semidirect
180. #556D157semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237580082
#237576652
#237576632
#237576552
#237574952
#237574922
#237574882
#237574852
#237574752
#237574732
#237574042
#237573832
#237573782
C4×#145789412
#556×C1572
#3401157